A321915 Tetrangle where T(n,H(u),H(v)) is the coefficient of h(v) in m(u), where u and v are integer partitions of n, H is Heinz number, m is monomial symmetric functions, and h is homogeneous symmetric functions.

1, 2, -1, -1, 1, 3, -3, 1, -3, 5, -2, 1, -2, 1, 4, -2, -4, 4, -1, -2, 3, 2, -4, 1, -4, 2, 7, -7, 2, 4, -4, -7, 10, -3, -1, 1, 2, -3, 1, 5, -5, -5, 5, 5, -5, 1, -5, 9, 5, -7, -9, 9, -2, -5, 5, 11, -11, -8, 10, -2, 5, -7, -11, 14, 10, -14, 3, 5, -9, -8, 10, 12

Offset: 1

Author

Gus Wiseman, Nov 22 2018

Keywords

• sign
• tabf

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Also the coefficient of e(v) in f(u), where f is forgotten symmetric functions and e is elementary symmetric functions.

Examples

			Tetrangle begins:
  (1):  1
.
  (2):   2 -1
  (11): -1  1
.
  (3):    3 -3  1
  (21):  -3  5 -2
  (111):  1 -2  1
.
  (4):     4 -2 -4  4 -1
  (22):   -2  3  2 -4  1
  (31):   -4  2  7 -7  2
  (211):   4 -4 -7 10 -3
  (1111): -1  1  2 -3  1
.
  (5):      5 -5 -5  5  5 -5  1
  (41):    -5  9  5 -7 -9  9 -2
  (32):    -5  5 11 11 -8 10 -2
  (221):    5 -7 11 14 10 14  3
  (311):    5 -9 -8 10 12 13  3
  (2111):  -5  9 10 14 13 17 -4
  (11111):  1 -2 -2  3  3 -4  1
For example, row 14 gives: m(32) = -5h(5) + 11h(32) + 5h(41) - 11h(221) - 8h(311) + 10h(2111) - 2h(11111).
		

Links

• Wikipedia, Symmetric polynomial

Crossrefs

This is a regrouping of the triangle A321748. Row sums are A155972.

Cf. A005651, A008480, A056239, A124794, A124795, A215366, A318284, A319191, A319193, A321912-A321935.